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stilyo
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Posts: 16
Joined: January 12th, 2009, 6:31 pm

Inverse Relationship

June 1st, 2022, 1:04 am

I have a dependent variable Y(t) which depends on an independent variable X(t) in the following way:
Y(t) = p_0*X(t)+p_1*X(t-1)+p_2*X(t-2)+...+p_n*X(t-n) where p_i=cosnt, 0<=p_i<=1 and sum(p_0+...+p_n)=1. So, given a history of n consecutive variables X(t), I can always calculate the "current" Y(t).

I want to do the opposite as well, given Y(t), Y(t-1),...., Y(t-n), calculate directly X(t). I could do it with brute force or maybe even with a pen and paper by writing out some monstrous algebraic equation but is there a neat way to invert this thing and solve for X(t)? Thanks a lot for any ideas!
 
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Alan
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Joined: December 19th, 2001, 4:01 am
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Re: Inverse Relationship

June 1st, 2022, 2:27 am

Assume X(t) = 0 for t < 0, and start with Y(0) = p_0 X(0) which is trivial to invert. Then just work sequentially through Y(1) , Y(2), etc.
 
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stilyo
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Joined: January 12th, 2009, 6:31 pm

Re: Inverse Relationship

June 4th, 2022, 2:09 am

Assume X(t) = 0 for t < 0, and start with Y(0) = p_0 X(0) which is trivial to invert. Then just work sequentially through Y(1) , Y(2), etc.
Thanks, Alan. Interesting idea leading to simplifications but I don’t think I can assume that. T=0 is today and I have a sequence of well-defined, non-zero X(t), Y(t) stretching back in time.
 
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Alan
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Re: Inverse Relationship

June 5th, 2022, 3:07 am

Stretching back forever? I don't believe it. There was a big bang you know.   :D